本文選題：耦合非線性方程組 + 改進(jìn)的三次B-樣條函數(shù)��； 參考：《中國(guó)礦業(yè)大學(xué)》2017年碩士論文

【摘要】：本篇論文主要研究?jī)深?lèi)耦合的非線性偏微分方程組:廣義Zakharov方程組和Klein-Gordon-Zakharov方程組的Dirichlet初邊值問(wèn)題的數(shù)值解法。在這里,我們采用高精度的微分求積法求解上述兩類(lèi)偏微分方程組,在微分求積法中應(yīng)用改進(jìn)的三次B-樣條函數(shù)確定加權(quán)系數(shù)。于是偏微分方程組轉(zhuǎn)化為常微分方程組系統(tǒng),最終我們使用最優(yōu)四階,保時(shí)間步長(zhǎng)三階的強(qiáng)穩(wěn)定性的龍格庫(kù)塔法求解這個(gè)系統(tǒng)。緒論部分簡(jiǎn)單的介紹了廣義Zakharov方程組和Klein-Gordon-Zakharov方程組的研究背景以及研究?jī)?nèi)容和一些預(yù)備知識(shí)。本文中我們將微分求積法和改進(jìn)的三次B-樣條函數(shù)結(jié)合起來(lái)應(yīng)用在二維、三維的廣義Zakharov方程組中求解該方程組的數(shù)值解。同時(shí)我們繼續(xù)使用改進(jìn)的三次B-樣條函數(shù)微分求積法求解Klein-Gordon-Zakharov方程組。緊接著我們對(duì)這兩個(gè)方程組進(jìn)行了數(shù)值模擬,將本文提出的數(shù)值方法和已有的研究這兩類(lèi)方程組的有限差分法得到的數(shù)值解與精確解進(jìn)行比較。從結(jié)果可以看出,與有限差分法相比,我們用改進(jìn)的三次B-樣條函數(shù)微分求積法得到的數(shù)值解更加地接近于精確解,即誤差也相對(duì)的較小。同時(shí)我們也繪出了兩個(gè)方程組的數(shù)值解與精確解的圖形,從圖形可直觀的看出由我們的方法得到的數(shù)值解圖形與精確解的圖形吻合的很好。尤其對(duì)于Klein-Gordon-Zakharov方程組,我們也模擬出了單個(gè)波的傳播過(guò)程以及在三維的情況下的數(shù)值解與精確解的圖形。從數(shù)值實(shí)驗(yàn)的結(jié)果可以看出我們的方法的有效性,以及與差分法相比,可得出我們的方法的準(zhǔn)確性。最后我們對(duì)本篇論文進(jìn)行了總結(jié)。
[Abstract]:In this paper, we study two kinds of coupled nonlinear partial differential equations: generalized Zakharov equations and Dirichlet initial-boundary value problems for Klein-Gordon-Zakharov equations. Here, we use the high-precision differential quadrature method to solve the above two kinds of partial differential equations, and apply the improved cubic B-spline function to determine the weighting coefficient in the differential quadrature method. So the system of partial differential equations is transformed into a system of ordinary differential equations. Finally, we use the Runge-Kutta method, which is an optimal fourth-order and three-order preserving time step, to solve the system. The introduction briefly introduces the research background of generalized Zakharov equations and Klein-Gordon-Zakharov equations, as well as the research contents and some preliminary knowledge. In this paper, the differential quadrature method and the improved cubic B-spline function are combined to solve the numerical solutions of the equations in two-dimensional and three-dimensional generalized Zakharov equations. At the same time, we continue to use the improved cubic B-spline function differential quadrature method to solve the Klein-Gordon-Zakharov equations. Then we carry on the numerical simulation to these two equations, and compare the numerical solution with the exact solution obtained by the numerical method proposed in this paper and the finite difference method which has been used to study these two kinds of equations. It can be seen from the results that compared with the finite difference method, the numerical solution obtained by using the improved cubic B-spline differential quadrature method is closer to the exact solution, that is, the error is also relatively small. At the same time, we also draw the figure of the numerical solution and the exact solution of the two equations. From the figure, we can directly see that the figure of the numerical solution obtained by our method is in good agreement with the figure of the exact solution. Especially for Klein-Gordon-Zakharov equations, we also simulate the propagation process of single wave and the figure of numerical solution and exact solution in three dimensional case. The results of numerical experiments show the validity of our method and the accuracy of our method compared with the difference method. Finally, we summarize this paper.
【學(xué)位授予單位】：中國(guó)礦業(yè)大學(xué)
【學(xué)位級(jí)別】：碩士
【學(xué)位授予年份】：2017
【分類(lèi)號(hào)】：O241.82

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